Starlight

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

Assumes Opacity and Asteroseismology.

17 min read 9 figures Light is the only messenger Measuring the unreachable

For about thirty years the Sun was the best-tested object in astrophysics. A model built from its measured surface composition, with standard nuclear physics and standard opacities, reproduced the sound speed that its own oscillations gave to about a part in a thousand at every depth. Nothing else in stellar astrophysics was checked that well, and the agreement was quoted as the reason to believe everything else that used the same machinery.

Then somebody re-measured the composition.

$A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.$

Fig. 1 The fractional difference between the sound speed the Sun’s oscillations measure and the sound speed a model predicts, against fractional radius. Zero is agreement. The lower curve is the model built on the abundances used until the mid-2000s; it hugs the axis. The upper curve is the same model with abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard — and the disagreement grows to nearly one per cent, peaking just beneath the base of the convection zone.

What was re-measured, and why it was better

A solar abundance is obtained by measuring the strength of an absorption line and asking what quantity of the element would produce it. The chain from one to the other passes through a model of the atmosphere, and until the 2000s that model was one-dimensional: a stack of plane-parallel layers in hydrostatic equilibrium, with convection represented by a mixing-length parameter, and with the populations of atomic levels set by local thermodynamic equilibrium.

Every one of those assumptions is known to be wrong in detail. The solar photosphere is not plane-parallel; it is granulated, with hot rising columns and cool sinking lanes at different temperatures and different velocities, and a line forms across all of them. It is not in local thermodynamic equilibrium either: at the densities where the weak lines form, radiative rates compete with collisional ones and the level populations depart from their Boltzmann values.

The re-measurement replaced both. Three-dimensional radiation-hydrodynamic simulations produce a granulation pattern that reproduces the observed line shapes — the asymmetries and the wavelength shifts that a one-dimensional model cannot make at all — and the departures from equilibrium were computed rather than assumed. The result was a set of abundances in which carbon, nitrogen and oxygen fell by roughly thirty per cent. Nobody has produced a serious argument that the older abundances were the better measurement. The new ones reproduce line shapes the old ones could not, they agree between different lines of the same element far better, and they agree with the composition of the solar wind and of meteorites where those can be compared — the same meteoritic scale that anchors every abundance ratio in galactic chemical evolution. The problem is not that the measurement is doubted. The problem is what it does downstream.

It is worth being clear about what “thirty per cent” refers to, because abundance notation is a trap for the unwary. Solar abundances are quoted logarithmically on a scale where hydrogen is 12, so oxygen moving from 8.83 to 8.66 is a factor of 1.5 in number. The overall metallicity $Z$ , the mass fraction in everything heavier than helium, moved from about 0.0195 to about 0.0134 — from two per cent of the Sun to one and a third. Those are the two numbers that matter downstream, and the second is what a model actually consumes.

The revision also changed the ratios rather than merely the scale. Carbon and oxygen fell furthest; iron barely moved, because iron’s abundance is measured from many lines including some relatively insensitive to the atmosphere, and it has been stable for decades. That detail matters below, because iron and oxygen contribute to the opacity at different temperatures, and a resolution that adjusts one of them is not the same as a resolution that adjusts both.

Why a surface abundance changes an interior

The abundances of carbon, nitrogen and oxygen are a few parts in a thousand by mass. It is not obvious that reducing them by thirty per cent should change anything at all about the middle of the Sun.

They matter because they are opacity. At the temperatures just below the convection zone — two to three million kelvin — hydrogen and helium are fully ionised and contribute almost nothing to the bound-free absorption; what absorbs is the partially ionised heavier elements, and oxygen and iron dominate. The Rosseland mean is a harmonic mean, so it is set by the frequencies at which the gas is most transparent, and the heavy elements are what fill in those windows.

$Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 10,343 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,400 K, where hydrogen is 78% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.$

Fig. 2 The three contributions, against temperature. Around a few million kelvin the bound-free term dominates, and it is proportional to the metal abundance directly. Remove thirty per cent of the metals and the opacity there falls by something like fifteen per cent, which changes how steep the radiative temperature gradient has to be to carry the flux — and therefore where that gradient exceeds the adiabatic one and convection sets in.

$Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.2865, Z = 0.0135, on logarithmic axes, at 10⁻⁷ g/cm³ and 10⁻⁶ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 7,118 cm²/g at 5,800 K, against the 0.40 drawn here. The peak sits at 15,100 K, where hydrogen is 75% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.$

Fig. 3 The same three contributions with the metals cut by a third —

Z = 0.0135

against

0.02

, which is roughly the size of the revision that started all this, with the difference put into helium so the composition still sums to one. Electron scattering does not move, because it does not care what the electrons came off. The negative-hydrogen term barely moves. The bound-free curve drops by very nearly the factor the metals dropped by, because it is proportional to them, and around a few million kelvin it is what the total is made of. That single proportionality is the whole mechanism by which a spectroscopic measurement of the surface reaches the base of the convection zone.

Lower opacity means shallower convection. The model’s convection-zone base moves from 0.715 of the radius to 0.729, against a seismic value of $0.713 \pm 0.001$ — a discrepancy of sixteen standard deviations in a quantity that is measured better than almost anything else about the Sun.

There is a chain of three steps there and each is worth naming. Opacity sets how steep a temperature gradient is needed to carry the luminosity radiatively. Where that gradient exceeds the adiabatic one, the layer becomes unstable and convection takes over. So the boundary between the radiative interior and the convective envelope is a place where an opacity is compared with a thermodynamic derivative, and moving the opacity moves the boundary.

And the sound speed follows. The sound speed depends on the temperature and the mean molecular weight, both of which shift when the boundary moves and when the thermal structure below it adjusts, and the resulting mismatch peaks exactly where the boundary is.

The shape of the disagreement is the evidence

A disagreement of one part in a hundred could in principle mean many things. What makes opacity the suspect is where the disagreement sits.

It is not spread through the star. It is localised in a region a tenth of the radius wide just beneath the convection zone, which is exactly where the metals’ contribution to the opacity is largest and where the boundary itself lies. A problem with the nuclear reaction rates would show at the centre. A problem with the equation of state would show everywhere. A problem with the mixing length would show above the boundary, not below it.

The size required is also specific. Increasing the opacity by ten to fifteen per cent near the base of the convection zone, tapering to nothing above and below, restores both the boundary depth and the sound speed. That is a very particular prescription, and the fact that one adjustment fixes several independent discrepancies at once is the strongest argument that it is the right adjustment. There is a symmetry worth noticing in the way the two disciplines meet here. The spectroscopists measure a composition at the surface and infer nothing about the interior; the seismologists measure a structure in the interior and infer nothing about the composition. Neither measurement has changed. What has changed is the theory that connects them, and the connection runs entirely through the opacity — so a disagreement between them is, almost by construction, a statement about opacity or about nothing.

That is a rare and rather enviable position for a discrepancy to be in. Most disagreements in astrophysics have many candidate causes, and narrowing them is the work. This one had its candidate identified within months, and the twenty years since have been spent trying to test it.

The other numbers that broke with it

The sound speed is the most quoted symptom and it is not the only one.

The convection zone depth, as above: 0.713 measured against 0.729 modelled.

The surface helium abundance, measured seismically from the signature of the second helium ionisation zone: 0.2485 measured against about 0.229 in the new-abundance model. Helium is not measurable in the solar spectrum at all, so this is a genuinely independent constraint, and it fails in the same direction.

The density profile, inverted separately from the sound speed by the same construction of localised averages, which shows a discrepancy of the same size and location.

The solar neutrino fluxes from the CNO cycle, which are directly proportional to the carbon and nitrogen abundances in the core. These were measured for the first time in 2020, by a detector deep under a mountain — a descendant of the experiments that found the only thing that leaves a stellar centre going missing, and the value came out closer to the older, higher abundances than to the newer ones — though with error bars wide enough that the result is suggestive rather than decisive.

Fig. 4 Why the surface is the wrong place to look for the problem. The photosphere is a layer a few hundred kilometres thick on a body seven hundred thousand kilometres across, and the abundances are measured in it. What the seismology disagrees about is half a million kilometres below, and the only thing connecting the two is the assumption that the convection zone is well mixed — which is the one assumption in this essay that nobody disputes.

Where the suspicion was tested

If the resolution is an opacity that is too low, the place to check is a laboratory.

Reproducing the conditions at the base of the solar convection zone means iron at about two million kelvin and an electron density of $10^{29}$ per cubic metre, held long enough and uniformly enough to measure a transmission spectrum. That was done at a large pulsed-power facility, using a Z-pinch to heat a thin iron sample and a separate burst of X-rays to probe it.

The measured iron opacity came out thirty to four hundred per cent higher than the theoretical values, depending on wavelength, with the largest discrepancies in the windows between lines — which is exactly where a harmonic mean is most sensitive. Folded into a solar model, the measured iron opacity supplies something like half of what is needed.

The result has not been reproduced independently, and the theoretical community has not identified what the calculations are missing. It is the strongest single piece of evidence in the direction the seismology points, and it is one experiment.

Fig. 5 Why an error in the windows matters more than an error in the lines. The Rosseland mean weights the inverse of the opacity, so it is dominated by the frequencies at which the gas is most transparent — a gap in the absorption lets flux through and lowers the mean far more than an extra line raises it. A calculation that gets the lines right and the gaps wrong will get the mean wrong in the direction the experiment found.

What makes the experiment hard is worth a sentence, because it explains why there is only one of them. The sample must be heated uniformly, held long enough to be in a known state, and probed by a source bright enough to measure transmission through it — all within a few nanoseconds, at conditions that exist naturally only inside stars. The facility used is one of a handful in the world, the shot rate is a few per day, and each measurement is a substantial fraction of a year’s programme. Independent reproduction means another facility deciding to spend that.

Why it cannot be tuned away

A reasonable first reaction is that a solar model has parameters, and that a disagreement of one part in a hundred ought to be absorbable by adjusting one of them. Setting out why that does not work is worth the space, because the answer says what kind of error the opacity would have to be.

A standard solar model has three adjustable quantities and no more: the initial helium mass fraction, the initial ratio of heavy elements to hydrogen, and the mixing-length parameter that stands in for convection. Everything else — the nuclear rates, the equation of state, the opacities, the treatment of gravitational settling — is fixed by physics computed somewhere else and imported.

Those three are already spent. The model is required to arrive, after four and a half billion years, at the observed luminosity, the observed radius, and the observed surface ratio of metals to hydrogen. Three conditions, three unknowns, one solution. Nothing is left over to spend on a sound speed.

So the sound-speed profile is not a fit. It is a prediction made by a model with no remaining freedom, and the same is true of the convection-zone depth and of the surface helium abundance. That is what licenses quoting the disagreement in standard deviations at all: there is no parameter whose adjustment would move it without breaking one of the three conditions that fixed the model in the first place.

Fig. 6 Why an opacity is hard to raise on purpose. The upper panel is a synthetic spectrum of a continuum with six lines per unit

x = h\nu/kT

laid over it, weighted by the Rosseland function that decides which frequencies carry the flux. The two horizontal lines are the two ways of averaging it, and they sit 75 times apart. The Planck mean is an ordinary average and lands among the lines; the Rosseland mean is a harmonic average of

1/\kappa

and therefore lands in the gaps, because radiation escapes where the gas is transparent. Adding lines to a spectrum raises the Planck mean in proportion and hardly moves the Rosseland mean at all — which is why “the opacities might be higher” is a much weaker statement than it sounds, and why the missing opacity has to come from the continuum or from lines dense enough to close the gaps.

It also constrains the shape of any repair, and this is the part that is usually skipped. Raising the opacity everywhere by fifteen per cent does not help, because the calibration reabsorbs it — the initial helium and the mixing length shift to restore the luminosity and the radius, and the structure returns to something close to where it began. What is required is a change in the opacity’s temperature dependence: more absorption near two million kelvin, and none at the centre, where the same increase would alter the nuclear burning and therefore the luminosity that has already been matched.

That is a far more specific demand than “the opacities are uncertain at the ten per cent level”, which is true and is not by itself a resolution. An uncertainty is a band drawn around a curve; what the seismology asks for is a bump in a particular place, of a particular width, with the rest of the curve left where it was. The iron experiment is interesting precisely because the excess it found was concentrated in the windows between lines at those temperatures — a bump of roughly the right shape, rather than an overall scale factor.

And it explains the shape of the alternatives below. Each of them is an attempt to supply a localised change by some other route — a composition that differs with depth, a mixing that alters the gradient, an element whose abundance is unconstrained — because localisation is the first condition any resolution has to meet.

The alternatives, and why none has taken hold

Several other resolutions have been proposed and each has a difficulty.

Accretion of metal-poor gas. If the young Sun accreted material from which the planets had already removed the heavy elements, its convection zone would be metal-poor relative to its interior, and the surface abundance would not represent the whole star. This works, and it requires the accretion to have happened at a particular epoch and in a particular amount, which makes it a fit rather than a prediction.

Fig. 7 And what halving the forest does. Three lines per unit

x

rather than six: the Planck mean falls with the line density, and the two means are now 67 times apart rather than 75. The gap between them narrows very slowly indeed, because closing it requires the lines to overlap rather than merely to be numerous — a forest twice as dense is still mostly gaps. That is the quantitative form of the demand the previous section arrived at, and it is why every proposal on this list is an attempt to change the continuum or the composition rather than to add lines.

Rotationally induced mixing. Extra mixing below the convection zone changes the composition gradient and the sound speed. It helps a little and not enough, and it makes the lithium depletion problem worse.

Enhanced neon. Neon’s abundance in the Sun cannot be measured spectroscopically — it has no photospheric lines — and is inferred from other stars via its ratio to oxygen. If solar neon were substantially higher than assumed, it would supply the missing opacity. Measurements in nearby stars and in the solar corona have gone both ways.

The abundances are wrong after all. A minority position, requiring that the three-dimensional non-equilibrium analyses share a common systematic. Independent groups have now done the analysis with different codes and reached similar answers, which weakens this considerably.

Fig. 8 The same opacity read as a slope rather than a value, at three densities two decades apart.

\partial\ln\kappa/\partial\ln T

is what decides whether a layer damps a disturbance or amplifies it, and the window where it is positive — the partial-ionisation zone — moves and narrows with density. The density dependence is the reason this cannot be settled by a single curve. The opacity of a gas is a function of two variables and every statement in this essay is a statement about one slice through it; a revision that helps at the base of the convection zone, where the density is a tenth of a gram per cubic centimetre, is a different revision from one that helps in the photosphere.

What the disagreement is worth

It is tempting to read a one-per-cent discrepancy in a sound speed as a technicality. It is not, for three reasons.

The first is that the Sun is the calibrator. Every stellar model in astrophysics — the ones that give ages to clusters, masses to exoplanet hosts, and distances from a main-sequence fit — uses opacity tables and a mixing length calibrated on the Sun. If the tables are wrong by fifteen per cent in a regime that matters, the error propagates into every one of those, and not by a fixed factor.

The second is that this is what a real systematic error looks like from the inside. Two measurements, each careful, each improved, disagreeing at ten times their quoted errors; a suspect that is a third quantity neither of them measured; and a laboratory test that supports the suspicion without settling it. The Sun is the object best positioned to be checked this way, and it has taken twenty years without resolution, on a body whose interior was once quoted as settled — which is a useful calibration on how much confidence to place in the interiors of stars nobody can measure at all.

There is a third reason, and it is the one that makes the problem tractable rather than merely irritating. The disagreement is between two measurements and not between a measurement and a taste: the sound-speed profile is inverted from millions of observed oscillation frequencies and the abundances are read off line profiles, so whichever of them is wrong is wrong in a way that some third measurement can reach. That is why the argument has moved to the laboratory and to the neutrinos rather than staying in the modelling, and it is the reason to expect an answer at all. One more reading of the same opacity table covers the densities a stellar envelope actually spans.

$Opacity against temperature, and the three things that supply it. The Rosseland mean opacity of a gas of composition X = 0.7, Y = 0.28, Z = 0.02, on logarithmic axes, at 10⁻⁵ g/cm³ and 10⁻⁴ g/cm³. The three faint curves are the separate processes at the first density — electron scattering, the Kramers bound-free and free-free term, and the negative hydrogen ion — and the solid curve is their sum. Every one of them is multiplied by the fraction of hydrogen the Saha equation says is ionised at that temperature and density, or by one minus it for H⁻, which is the only reason the low-temperature end is a picture of a star rather than of a formula outside its range: ungated, Kramers alone gives 1,034,309 cm²/g at 5,800 K, against the 17.63 drawn here. The peak sits at 21,800 K, where hydrogen is 61% ionised — that bump is not a detail, it is the engine of a Cepheid — and the flat floor at high temperature is electron scattering, which is the one term with no temperature in it at all.$

Fig. 9 The three contributions at densities two orders of magnitude above a photosphere’s. The bound–free term rises with density and the electron-scattering floor does not, so the region in which a revision to the bound–free opacity changes a stellar model is the deep envelope rather than the surface.

Where the ladder goes

The earlier rungs of this anchor established what an opacity is and how the averaging works: that the photosphere is a depth rather than a surface, and that the mean is dominated by the gaps. This one is about what happens when the number is wrong.

The next steps go in two directions. One is experimental: the iron measurement needs to be repeated at other facilities, and extended to the other elements that matter — oxygen, neon, chromium, nickel. The other is observational: the CNO neutrino measurement will improve, and it is the only probe of the core’s composition that does not pass through a model atmosphere. If it settles on the low abundances, opacity is the answer; if on the high ones, something about the surface of the Sun is not representative of the whole.